Who owns your face? Believe it or not, the answer depends on which state you live in, and chances are, you live in one that hasn’t even weighed in yet.

That could soon change. For the fourth time this year, Facebook Inc. was hit with a class-action lawsuit by an Illinois resident who says its face-recognition software violates an unusual state privacy law there. The latest complaint, filed Monday, underscores a quiet but high-stakes legal battle for the social networking giant, one that could reverberate throughout the rest of the U.S. tech industry and much of the private sector.